Algebraic techniques 2 - Cambridge University Pressassets.cambridge.org/97811076/71812/excerpt/9781107671812_excerpt.pdf6 Chapter 1 Algebraic techniques 2 and indices Example 1 Using

Chapter 1 Algebraic techniques 2 and indices2

Chapter

Algebraic techniques 2 and indices1

What you will learn 1A The language of algebra REVISION

1B Substitution and equivalence 1C Adding and subtracting terms REVISION

1D Multiplying and dividing terms REVISION

1E Adding and subtracting algebraic fractions EXTENSION

1F Multiplying and dividing algebraic fractions EXTENSION

1G Expanding brackets 1H Factorising expressions 1I Applying algebra 1J Index laws for multiplication and division 1K The zero index and power of a power

Cambridge University Press978-1-107-67181-2 - CambridgeMATHS: NSW Syllabus for the Australian Curriculum: Year 8Stuart Palmer, David Greenwood, Bryn Humberstone, Justin Robinson, Jenny Goodman and Jennifer VaughanExcerptMore information

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Number and Algebra 3

Avatar algebra

NSW Syllabus for the Australian CurriculumStrand: Number and Algebra

Substrand: ALGEBRAIC TECHNIQUES

Outcomes

A student generalises number properties to operate with algebraic expressions.

(MA4–8NA)

A student operates with positive-integer and zero indices of numerical bases.

(MA4–9NA)

3

Computer gaming is a billion-dollar industry that employs a range of computer specialists including game programmers. To create a virtual three-dimensional world on a two-dimensional screen requires the continual variation of thousands of numbers (or coordinates); for example, if an avatar leaps up, the position of its shadow must be changed. In the two-dimensional image, the change in any one measurement results in many, many other measurement changes in order to produce a realistic image. It would be annoying for programmers if every time they changed one measurement they had to write a separate program instruction for hundreds of changes! If their avatar jumps into a doorway, the door’s dimensions, the light and shadows, the size and movement of enemies, the viewing angle etc. must all be recalculated. However, a programmer avoids such tedious work by using algebra. For example, an algebraic rule making a door height equal twice the avatar’s height can be written into a game’s program.

Algebraic rules linking many varying but related quantities are programmed into computer games. Other examples of related variables include how fast the avatar runs and how fast the background goes past, the avatar’s direction of movement and the route that its enemies follow. Computer-game programmers deal with many more complex and sophisticated issues of input and output. Avatars, programmed with artifi cial intelligence, can make decisions, react unpredictably, interact with the terrain and try to outwit their human enemy. Expertise in mathematics, physics, logic, problem-solving and attention to detail are all essential skills for the creation of realistic and exciting computer games.






Chapter 1 Algebraic techniques 2 and indices4 Chapter 1 Algebraic techniques 2 and indices4

1 Evaluate:

a 8 + 4 × 6 b 4 × 5 − 2 × 3

c 12 − (6 + 2) + 8 d 3(6 + 4)

2 Evaluate:

a 10 + 6 − 12 b 4 − 7

c −3 − 8 d −1 − 1 − 1

3 Write an expression for:

a 5 more than x b 7 less than m

c the product of x and y d half of w

e double the sum of p and q

4 If y = 2x + 5, fi nd the value of y when x = 1.2.

5 Complete the tables using the given equations.

a M = 2A + 3

A 0 3 7 10

M

b y = 1

2(x + 1)

x 1 3 11 0

y

6 Substitute x = 6 and y = −2 into each expression and then evaluate.

a x + y b xy c 3x − y d 2x + 3y

7 Write these numbers in expanded form.

a 52 b 24 c 33 d (−8)3

8 Evaluate:

a 53 b 42 + 52 c (−4)2 d 33 − 22

9 Complete:

a 5 + 5 + 5 + 5 = 4 × b 6 + 6 + 6 + 6 + 6 + 6 = × 6

c 10 ÷ 5 = 5

d 9 × 9 × 9 × 8 × 8 = 9 × 8

e 117 × 21 = 117 × 20 + 117 ×

10 Write down the HCF (highest common factor) of:

a 24 and 36 b 15 and 36 c 48 and 96

Pre-

test







The language of algebraA pronumeral is a letter that can represent one or more

numbers. For instance, x could represent the number of

goals a particular soccer player scored last year. Or p could

represent the price (in dollars) of a book. If a pronumeral can

take different values it is also called a variable.

Let’s start: Algebra sortConsider the four expressions x + 2, x × 2, x − 2

and x ÷ 2.

• If you know that x is 10, sort the four values

from lowest to highest.

• Give an example of a value of x that would

make x × 2 less than x + 2.

• Try different values of x to see if you can:

– make x ÷ 2 less than x − 2

– make x + 2 less than x − 2

– make x × 2 less than x ÷ 2

1A

This soccer player scored x goals last year.x goals last year.x

REVISION

� In algebra, a letter can be used to represent one or more numbers. These letters are called

pronumerals. A variable is a letter that is used to represent more than one number.

� a × b is written ab and a ÷ b is written a

b.

� a × a is written a2.

� An expression is a combination of numbers and pronumerals combined with mathematical

operations, e.g. 3x + 2yz and 8 ÷ (3a − 2b) + 41 are expressions.

� A term is a part of an expression with only pronumerals, numbers, multiplication and division,

e.g. 9a, 10cd and 3

5

x are all terms.

� A coeffi cient is the number in front of a pronumeral. If the term is being subtracted, the coeffi cient

is a negative number, and if there is no number in front, the coeffi cient is 1. For

the expression 3x + y − 7z, the coeffi cient of x is 3, the coeffi cient of y is 1 and the coeffi cient

of z is −7.

� A term that does not contain any variables is called a constant term.

� The sum of a and b is a + b.

� The difference of a and b is a − b.

� The product of a and b is a × b.

� The quotient of a and b is a ÷ b.

� The square of a is a2.

Key

idea

s







Example 1 Using the language of algebra

a List the individual terms in the expression 4a + b − 12c + 5.

b In the expression 4a + b − 12c + 5 state the coeffi cients of a, b, c and d.

c What is the constant term in 4a + b − 12c + 5?

d State the coeffi cient of b in the expression 3a + 4ab + 5b2 + 7b.

SOLUTION EXPLANATION

a There are four terms: 4a, b, 12c and 5. Each part of an expression is a term. Terms get added

(or subtracted) to make an expression.

b The coeffi cient of a is 4.

The coeffi cient of bThe coeffi cient of bThe coeffi cient of is 1.

The coeffi cient of The coeffi cient of c is is −12.12.

The coeffi cient of d is 0.d is 0.d

The coeffi cient is the number in front of a pronumeral.

For b the coeffi cient is 1 because b is the same as 1 × b.

For For c the coeffi cient is the coeffi cient is −12 because this term is being 12 because this term is being

subtracted. For d the coeffi cient is 0 because there are d the coeffi cient is 0 because there are d

no terms with d.

c 5 A constant term is any term that does not contain a

pronumeral.

d 7 Although there is a 4 in front of ab and a 5 in front of

b2, neither of these is a term containing just b, so they

should be ignored.

Example 2 Creating expressions from a description

Write an expression for each of the following.

a The sum of 3 and k b The product of m and 7

c 5 is added to one half of k d The sum of a and b is doubled


a 3 + k The word ‘sum’ means +.

b m × 7 or 7m The word ‘product’ means ×.

c 1

25k + or

k

25+ One half of k can be written k can be written k

1

2× k (because ‘of’

means ×), or k

22 because k is being divided by two.k is being divided by two.k

d (a + b) × 2 or 2(a + b) The values of a and b are being added and the result is

multiplied by 2. Grouping symbols (the brackets) are

required to multiply the whole result by two and not

just the value of b.







WORKING

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Exercise 1A R E V I S I O N

1 The expression 3a + 2b + 5c has three terms.

a List the terms.

b State the coeffi cient of:

i a

ii b

iii c

c Write another expression with three terms.

2 The expression 5a + 7b + c − 3ab + 6 has fi ve terms.

a State the constant term.

b State the coeffi cient of:

i a

ii b

iii c

c Write another expression that has fi ve terms.

3 Match each of the following worded statements with the correct mathematical expression.

a The sum of x and 7 A 3 − x

b 3 less than x B x

3c x is divided by 2 C x − 3

d x is tripled D 3x

e x is subtracted from 3 E x

2f x is divided by 3 F x + 7

Example 1a,b

Example 1c,d

Example 2a,b

WORKING

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i state how many terms there are ii list the terms

a 7a + 2b + c b 19y − 52x + 32

c a + 2b d 7u − 3v + 2a + 123c

e 10f + 2be f 9 − 2b + 4c + d + e

g 5 − x2y + 4abc − 2nk h ab + 2bc + 3cd + 4de

5 For each of the following expressions, state the coeffi cient of b.

a 3a + 2b + c b 3a + b + 2c

c 4a + 9b + 2c + d d 3a − 2b + f

e b + 2a + 4 f 2a + 5c

g 7 − 54c + d h 5a − 6b + c

i 4a − b + c + d j 2a + 4b2 − 12b

k 7a − b + c l 8a + c − 3b + d







WORKING

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6 Write an expression for each of the following.

a 7 more than y b 3 less than x

c The sum of a and b d The product of 4 and p

e Half of q is subtracted from 4 f One third of r is added to 10

g The sum of b and c multiplied by 2 h The sum of b and twice the value of c

i The product of a, b and c divided by 7 j A quarter of a added to half of b

k The quotient of x and 2y l The difference of a and half of b

m The product of k and itself n The square of w

7 Describe each of the following expressions in words.

a 3 + x b a + b c 4 × b × c

d 2a + b e (4 − b) × 2 f 4 − 2b

Example 2c,d

WORKING

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R PSC8 Marcela buys 7 plants from the local nursery.

a If the cost is $x for each plant, write an

expression for the total cost in dollars.

b If the cost of each plant is decreased by

$3 during a sale, write an expression for:

i the new cost per plant in dollars

ii the new total cost in dollars of the

7 plants

9 Francine earns $p per week for her job. She

works for 48 weeks each year. Write an

expression for the amount she earns:

a in a fortnight

b in one year

c in one year if her wage is increased

by $20 per week after she has already

worked 30 weeks in the year

10 Jon likes to purchase DVDs of some TV

shows. One show, Numbers, costs $a per

season, and another show, Proof by Induction,

costs $b per season. Write an expression for the cost of:

a 4 seasons of Numbers

b 7 seasons of Proof by Induction

c 5 seasons of both shows

d all 7 seasons of each show, if the fi nal price is halved when purchased in a sale

11 A plumber charges a $70 call-out fee and then $90 per hour. Write an expression for the

total cost of calling a plumber out for x hours.

1A







12 A mobile phone call costs 20 cents connection fee and then 50 cents per minute.

a Write an expression for the total cost (in cents) of a call lasting t minutes.

b Write an expression for the total cost (in dollars) of a call lasting t minutes.

c Write an expression for the total cost (in dollars) of a call lasting t hours.

WORKING

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Enrichment: Algebraic alphabet

16 An expression contains 26 terms, one for each letter of the alphabet. It starts

a + 4b + 9c + 16d + 25e + …

a What is the coefficient of f ?

b What is the coefficient of z?

c Which pronumeral has a coefficient of 400?

d One term is removed and now the coefficient of k is zero. What was the term?

e Another expression containing 26 terms starts a + 2b + 4c + 8d + 16e + … What is the sum of all

the coefficients?

WORKING

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13 If x is a positive number, classify the following statements as true or false.

a x is always smaller than 2 × x.

b x is always smaller than x + 2.

c x is always smaller than x2.

d 1 − x is always less than 4 − x.

e x − 3 is always a positive number.

f x + x − 1 is always a positive number.

14 If b is a negative number, classify the following statements as true or false. Give a brief reason.

a b − 4 must be negative.

b b + 2 could be negative.

c b × 2 could be positive.

d b + b must be negative.

15 What is the difference between 2a + 5 and 2(a + 5)? Give an expression in words to describe each of

them and describe how the grouping symbols change the meaning.







Substitution and equivalenceOne common thing to do with algebraic expressions is to replace the pronumerals (also known as

variables) with numbers. This is referred to as substitution, or evaluation. In the expression 4 + x we can

substitute x = 3 to get the result 7. Two expressions are called equivalent if they always give the same

result when a number is substituted. For example, 4 + x and x + 4 are equivalent, because no matter what

the value of x, 4 + x and x + 4 will be equal numbers.

Let’s start: AFL algebraIn Australian Rules football, the fi nal team score is given by 6x + y, where x is the number of goals and

y is the number of behinds scored.

• State the score if x = 3 and y = 4.

• If the score is 29, what are the values of x and y? Try to list all the possibilities.

• If y = 9 and the score is a two-digit number, what are the possible values of x?

1B

� To evaluate an expression or to substitute values means to replace each pronumeral in an

expression with a number to obtain a fi nal value.

For example, if a = 3 and b = 4, then we can evaluate the expression 7a + 2b + 5:

a b7 2a b7 2a b 5 7(3) 2(4) 5

21 8 5

34

+ +a b+ +a ba b7 2a b+ +a b7 2a b = +5 7= +5 7(3= +(3) 2= +) 2 ) 5+) 5

= +21= +21 8 5+8 5

=� Two expressions are equivalent if they have equal values regardless of the number that is

substituted for each pronumeral. The laws of arithmetic help to determine equivalence.

� The commutative laws of arithmetic tell us that a + b = b + a and a × b = b × a for any values of

a and b.

� The associative laws of arithmetic tell us that a + (b + c) = (a + b) + c and

a × (b × c) = (a × b) × c for any values of a and b.

Key

idea

s

Example 3 Substituting values

Substitute x = 3 and y = 6 to evaluate the following expressions.

a 5x b 5x2 + 2y + x


a 5 5

15

5 5x5 55 5=5 5

=( )3( )3 Remember that 5(3) is another way of writing 5 × 3.

b 5 2 5 3 2 6

5 9 12 3

45 12 3

6

2 25 22 25 2 5 32 25 3x y5 2x y5 2 x+ +5 2+ +5 22 2+ +2 25 22 25 2+ +5 22 25 2x y+ +x y5 2x y5 2+ +5 2x y5 2 = +2 2= +2 2= +5 3= +5 32 2= +2 25 32 25 3= +5 32 25 3 += +5 9= +5 9 += +45= +45 +=

2 2( )2 25 32 25 3( )5 32 25 3( )5 3( )5 3= +( )= +5 3= +5 3( )5 3= +5 32 2= +2 2( )2 2= +2 25 32 25 3= +5 32 25 3( )5 32 25 3= +5 32 25 3 ( )2 6( )2 6 ( )3( )3

( )5 9( )5 9( )5 9( )5 9= +( )= +5 9= +5 9( )5 9= +5 9

000

Replace all the pronumerals by their values

and remember the order in which to evaluate

(multiplication before addition).







Example 4 Deciding if expressions are equivalent

a Are x − 3 and 3 − x equivalent expressions?x equivalent expressions?x

b Are a + b and b + 2a − a equivalent expressions?


a No. The two expressions are equal if x = 3 (both equal zero).

But if x = 7 then x − 3 = 4 and 3 − x = −4.

Because they are not equal for every single value of x, they are not

equivalent.

b Yes. Regardless of the values of a and b substituted, the two

expressions are equal. It is not possible to check every single

number but we can check a few to be reasonably sure they are

equivalent.

For instance, if a = 3 and b = 5, then a + b = 8 and b + 2a − a = 8.

If a = 17 and b = −2 then a + b = 15 and b + 2a − a = 15.

Exercise 1B WORKING

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2 What is the result of evaluating 20 − b if b is equal to 12?

3 What is the value of a + 2b if a and b both equal 10?

4 a State the value of 4 + 2x if x = 5.

b State the value of 40 − 2x if x = 5.

c Are 4 + 2x and 40 − 2x equivalent expressions?

WORKING

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a 4 b 5 c 2 d 8

e 0 f −6 g −9 h −3

6 Substitute a = 4 and b = −3 into each of the following.

a 5a + 4 b 3b c a + b

d ab − 4 + b e 2 × (3a + 2b) f 100 − (10a + 10b)

g 12 6

a b+ h ab

b3

+ i 100

a ba b+a bj a2 + b k 5 × (b + 6)2 l a − 4b

Example 3a

Example 3b






Documents

Algebraic techniques 2 - Cambridge University Pressassets.cambridge.org/97811076/71812/excerpt/9781107671812_excerpt.pdf6 Chapter 1 Algebraic techniques 2 and indices Example 1 Using